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Simplifying (2x + -1)(2x + 1)(4x + 1) = (4x + -1) Reorder the terms: (-1 + 2x)(2x + 1)(4x + 1) = (4x + -1) Reorder the terms: (-1 + 2x)(1 + 2x)(4x + 1) = (4x + -1) Reorder the terms: (-1 + 2x)(1 + 2x)(1 + 4x) = (4x + -1) Multiply (-1 + 2x) * (1 + 2x) (-1(1 + 2x) + 2x * (1 + 2x))(1 + 4x) = (4x + -1) ((1 * -1 + 2x * -1) + 2x * (1 + 2x))(1 + 4x) = (4x + -1) ((-1 + -2x) + 2x * (1 + 2x))(1 + 4x) = (4x + -1) (-1 + -2x + (1 * 2x + 2x * 2x))(1 + 4x) = (4x + -1) (-1 + -2x + (2x + 4x2))(1 + 4x) = (4x + -1) Combine like terms: -2x + 2x = 0 (-1 + 0 + 4x2)(1 + 4x) = (4x + -1) (-1 + 4x2)(1 + 4x) = (4x + -1) Multiply (-1 + 4x2) * (1 + 4x) (-1(1 + 4x) + 4x2 * (1 + 4x)) = (4x + -1) ((1 * -1 + 4x * -1) + 4x2 * (1 + 4x)) = (4x + -1) ((-1 + -4x) + 4x2 * (1 + 4x)) = (4x + -1) (-1 + -4x + (1 * 4x2 + 4x * 4x2)) = (4x + -1) (-1 + -4x + (4x2 + 16x3)) = (4x + -1) (-1 + -4x + 4x2 + 16x3) = (4x + -1) Reorder the terms: -1 + -4x + 4x2 + 16x3 = (-1 + 4x) Remove parenthesis around (-1 + 4x) -1 + -4x + 4x2 + 16x3 = -1 + 4x Add '1' to each side of the equation. -1 + -4x + 4x2 + 1 + 16x3 = -1 + 1 + 4x Reorder the terms: -1 + 1 + -4x + 4x2 + 16x3 = -1 + 1 + 4x Combine like terms: -1 + 1 = 0 0 + -4x + 4x2 + 16x3 = -1 + 1 + 4x -4x + 4x2 + 16x3 = -1 + 1 + 4x Combine like terms: -1 + 1 = 0 -4x + 4x2 + 16x3 = 0 + 4x -4x + 4x2 + 16x3 = 4x Solving -4x + 4x2 + 16x3 = 4x Solving for variable 'x'. Reorder the terms: -4x + -4x + 4x2 + 16x3 = 4x + -4x Combine like terms: -4x + -4x = -8x -8x + 4x2 + 16x3 = 4x + -4x Combine like terms: 4x + -4x = 0 -8x + 4x2 + 16x3 = 0 Factor out the Greatest Common Factor (GCF), '4x'. 4x(-2 + x + 4x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(-2 + x + 4x2)' equal to zero and attempt to solve: Simplifying -2 + x + 4x2 = 0 Solving -2 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + 0.25x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.25x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.25x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.25x + x2 = 0 + 0.5 0.25x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.25x + x2 = 0.5 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.25x + 0.25 + x2 = 0.5 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + 0.25x + x2 = 0.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 0.5 = 0.866025404 Simplifying x + 0.5 = 0.866025404 Reorder the terms: 0.5 + x = 0.866025404 Solving 0.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.866025404 + -0.5 x = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 x = 0.366025404 Simplifying x = 0.366025404Subproblem 2
x + 0.5 = -0.866025404 Simplifying x + 0.5 = -0.866025404 Reorder the terms: 0.5 + x = -0.866025404 Solving 0.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.866025404 + -0.5 x = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 x = -1.366025404 Simplifying x = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.366025404, -1.366025404}Solution
x = {0, 0.366025404, -1.366025404}
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